For my project I needed some cheap clones of Nano and I bought them from DealExtreme. Four came, all without bootloader. I needed to program them with ArduinoISP. Three of them already works, fourth give me this error during bootloader upload:

If you're not getting a valid device signature (and 000000 is not at all valid), then you have a pretty significant problem. Since the same setup is working on other Nanos, perhaps that particular board is simply broken.

ArduinoISP is poor on error checking. In particular, the "avrdude: AVR device initialized and ready to accept instructions" doesn't mean anything, since ArduinoISP doesn't check the returned data from the "enter programming mode" instruction. The chip/board could be completely dead. (You'd probably get the same sequence with nothing connected...)

What sort of test equipment do you have to poke at things with?

Avoid using -F in avrdude. It's only purpose is to override conditions that you should be investigating and fixing, rather than overriding, and it can push error detection even further form the original error.